Section 9.1 Significance Tests: The Basics

[Pages:22]Section 9.1 Significance Tests: The Basics

? Confidence intervals are one of the two most common types of statistical inference.

? Use a confidence interval when your goal is to estimate a population parameter.

? The second common type of inference, called tests of significance, has a different goal: to assess the evidence provided by data about some claim concerning a population.

A significance test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess. The claim is a statement about a parameter, like the population proportion p or the population mean ?.

We express the results of a significance test in terms of a probability that measures how well the data and the claim agree.

? A basketball player claims to make 80% of the free throws that he attempts. We think he might be exaggerating. To test this claim, we'll ask him to shoot some free throws- virtually- using a simulation.

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? When do you have enough data to decide whether the player's claim is valid? How large a sample of shots do you need to make your decision?

Hypotheses

A significance test starts with a statement of the claims we want to compare

? They are the null, Ho, and alternative, Ha, hypotheses.

? The null hypothesis, Ho, is the claim tested by a statistical test. Often the null hypothesis is a statement of "no difference".

? The alternative hypothesis, Ha, is the claim about the population that we are trying to find evidence for.

In the free-throw shooter example, our hypotheses are

H0 : p = 0.80 Ha : p < 0.80 where p is the long-run proportion of made free throws.

In any significance test, the null hypothesis has the form

H0 : parameter = value

The alternative hypothesis has one of the forms

Ha : parameter < value Ha : parameter > value Ha : parameter value

One-sided alternative hypothesis

Two-sided alternative hypothesis

To determine the correct form of Ha, read the problem carefully. **Always state H0 and Ha in terms of population parameters

Anemia

Hemoglobin is a protein in red blood cells that carries oxygen from the lungs to body tissues. People with less than 12 grams of hemoglobin per deciliter of blood (g/dl) are anemic. A public health official in Jordan suspects that Jordanian children are at risk of anemia. He measures a random sample of 50 children.

a) Describe the parameter of interest in this setting.

The parameter of interest is the true mean ? amount of hemoglobin in Jordanian children.

b) State appropriate hypotheses for performing a significance test.

Because we are only concerned if Jordanian children have lower than 12 g/dl of hemoglobin, this will be one-sided. That is,

H0: ? = 12 Ha: ? < 12

? Looking at the basketball player example, he attempts 50 free-throws. He makes 32 of them. His sample proportion is 32/50 = 0.64

? We can simulate 400 sets of 50 shots assuming that the player is really an 80% shooter.

You can say how strong the evidence against the player's claim is by giving the probability that he would make as few as 32 out of 50 free throws if he really makes 80% in the long run.

The observed statistic is so unlikely if the actual parameter value is p = 0.80 that it gives convincing evidence that the player's claim is not true.

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